Four compartment trays with a measured set of colored counters. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
A joint probability has both an upper and a lower bound
For two events with marginal probabilities a and b, their intersection must lie between max(0,a+b−1) and min(a,b). The upper bound prevents an overlap from exceeding either event. The lower bound prevents the union from exceeding the whole sample space. Both conditions are needed to form a valid two-by-two table.
Check a high-prevalence example
If P(A) = 80% and P(B) = 70%, at least 50% must belong to both. An entered overlap of 20% would imply a union of 130%, which is impossible. An overlap of 60% is feasible: A-only is 20%, B-only is 10%, both is 60% and neither is 10%.
Independence is one feasible choice, not a default fact
Under independence, the intersection in that example would be 0.8×0.7 = 0.56. Another feasible overlap describes another dependence structure. Mutually exclusive events would require an overlap of zero, so that model is impossible for these large marginals. The Probability Calculator rejects the incompatible exclusive model rather than returning a union above 100%.
Repair the data source, not just the output
If rates do not form a feasible table, check whether they came from the same population, period and measurement definition. Rounding can create tiny discrepancies, but large contradictions should not be hidden by clipping arbitrary outputs into the probability range. Read inclusive OR versus exactly one after establishing a valid table. With three events, pairwise checks alone are no longer sufficient; the three-event calculator validates all eight disjoint regions.
Check the model behind your result
Use the Probability Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26